Global in time Strichartz inequalities on asymptotically flat manifolds with temperate trapping

Abstract

We prove global Strichartz inequalities for the Schrödinger equation on a large class of asymptotically conical manifolds. Letting $P$ be the nonnegative Laplace operator and $f_0\in C_0^{\infty }(\mathbb{R})$ be a smooth cutoff equal to $1$ near zero, we show first that the low frequency part of any solution $e^{-itP}{u}_0$, i.e. $f_0(P)e^{-itP}{u}_0$, enjoys the same global Strichartz estimates as on ${\mathbb{R}}^n$ in dimension $n\ge 3$. We also show that the high energy part $(1-f_0)(P)e^{-itP}{u}_0$ also satisfies global Strichartz estimates without loss of derivatives outside a compact set, even if the manifold has trapped geodesics but in a temperate sense. We then show that the full solution $e^{-itP}{u}_0$ satisfies global space-time Strichartz estimates if the trapped set is empty or sufficiently filamentary, and we derive a scattering theory for the $L^2$ critical nonlinear Schrödinger equation in this geometric framework.

80 pages

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